Lecture
Quantum neural networks (QNN) are computational neural network models based on the principles of quantum mechanics. The first ideas about quantum neural computation were independently published in 1995 by Subhash Kak and Ron Chrisley, who worked on the theory of quantum mind, which holds that quantum effects play a role in cognitive functions. However, typical research in QNNs involves combining classical artificial neural network models (which are widely used in machine learning for the important task of pattern recognition) with the advantages of quantum information to develop more efficient algorithms. One important motivation for this research is the difficulty of training classical neural networks, especially in big-data applications. It is hoped that features of quantum computing such as quantum parallelism, or interference and entanglement effects, can be exploited as resources. Since the technological realization of a quantum computer is still at an early stage, such quantum neural network models are for the most part theoretical proposals awaiting their full realization in physical experiments.
The idea of artificial quantum neural networks, first formulated in a paper, represents a combination of the concept of an artificial neural network and the paradigm of quantum computing. The first systematic treatment of artificial quantum neural
networks was given in the dissertation of T. Menneer (1998). Building on the work of Menneer, Narayanan, Coda, Matsui, Nishimura [35,36], Altaisky, Zhou and others, quantum-inspired neural network learning algorithms were constructed that are used at present
in various training programs and computer games [30,29]. The first scalable, hardware-implemented model of an artificial quantum neural network suitable for practical use was developed by D-Wave Systems Inc. and is a quantum Hopfield network built on superconducting quantum interference devices (SQUIDs). This paper analyzes the principles and possibilities of an alternative implementation of quantum neural networks based on quantum dots.
The possibility of using quantum neural-network algorithms in automated control systems, in associative memory devices, and in modeling biological and social networks is being considered
Research on quantum neural networks is still in its infancy, and numerous proposals and ideas of varying scale and mathematical rigor have been put forward. Most of them are based on the idea of replacing the classical binary or McCulloch-Pitts neuron with a qubit (which could be called a «quron»), resulting in nerve nodes that can be in a superposition of the «firing» and «resting' states.
Many proposals attempt to find a quantum equivalent for the perceptron unit from which neural networks are built. The problem is that non-linear activation functions do not readily fit the mathematical structure of quantum theory, since quantum evolution is described by linear operations and leads to a probabilistic observation. Ideas for emulating the perceptron's activation function using quantum-mechanical formalism range from ad hoc measurements to postulating non-linear quantum operators (a mathematical foundation that is disputed). A direct implementation of the activation function using the circuit model of quantum computation was recently proposed by Schuld, Sinayskiy, and Petruccione, based on the quantum phase estimation algorithm.
On a broader scale, researchers have attempted to generalize neural networks to a quantum setting. One way of constructing a quantum neuron is to first generalize the classical neurons, and then generalize them further to obtain unitary gates. Interactions between neurons can be controlled quantum-mechanically, via unitary gates, or classically, by measuring the states of the network. This high-level theoretical approach can be applied broadly, accommodating different types of networks and different implementations of quantum neurons, such as photonically implemented neurons and a quantum reservoir processor. Most learning algorithms follow the classical model of artificial-neural-network training to learn the input-output function of a given training set, and use classical feedback loops to update the parameters of the quantum system until they converge to an optimal configuration. Adiabatic models of quantum computation have also approached training as a parameter-optimization problem.
Quantum neural networks can be applied to algorithmic design: given qubits with tunable mutual interactions, one can attempt to learn the interactions by following the classical backpropagation rule, based on a training set of desired input-output relations taken to represent the desired behavior of the output algorithm. In this way, the quantum network «learns» the algorithm.
The quantum associative memory algorithm was introduced by Dan Ventura and Tony Martinez in 1999. The authors do not attempt to translate the structure of artificial neural network models into quantum theory, but instead propose a circuit-based algorithm for a quantum computer that emulates associative memory. Memory states (in Hopfield neural networks, stored in the weights of the neural connections) are written as a superposition, and a Grover-like quantum search algorithm retrieves the memory state closest to a given input. The advantage lies in the exponential storage capacity of the memory states; however, it remains an open question whether the model is meaningful with respect to the original purpose of Hopfield models, namely demonstrating how simplified artificial neural networks can model features of the brain.
Considerable interest has been generated by a «quantum» model that uses ideas from quantum theory to implement a neural network based on fuzzy logic.
In the near future, gate-based quantum computers will emerge with enough qubits and sufficient fidelity to run circuits and tasks that are out of reach for conventional computing machines. One approach to designing quantum algorithms — is to let the hardware architecture determine which set of gates to use.
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